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Non-maximal closed prime ideals in a unital commutative Banach algebra are accessible

Ramesh Garimella

TL;DR

The paper addresses whether non-maximal closed prime ideals in a unital commutative Banach algebra are accessible, proving that they must be the intersection of all closed ideals properly containing them. Using Zorn's Lemma and a detailed case analysis, it shows inaccessible closed prime ideals cannot occur, which yields automatic-continuity results: all derivations and epimorphisms on commutative unital semiprime Banach algebras are continuous, and the separating ideals associated with derivations or epimorphisms are nilpotent. Consequently, derivations on integral-domain and semiprime algebras are continuous, and epimorphisms onto integral domains are continuous as well. The work thus connects prime-ideal structure with automatic continuity phenomena in the Banach-algebra setting and highlights nilpotence of separating ideals as a recurring theme.

Abstract

It is proved that in a commutative unital Banach algebra, every non-maximal closed prime ideal is accessible. Specifically, it can be represented as the intersection of all closed ideals of the algebra that properly contain it. Consequently, all derivations and epimorphisms on commutative unital semi-prime Banach algebras are continuous. Moreover, any separating ideal in a commutative unital Banach algebra is nilpotent and, therefore, a nil ideal.

Non-maximal closed prime ideals in a unital commutative Banach algebra are accessible

TL;DR

The paper addresses whether non-maximal closed prime ideals in a unital commutative Banach algebra are accessible, proving that they must be the intersection of all closed ideals properly containing them. Using Zorn's Lemma and a detailed case analysis, it shows inaccessible closed prime ideals cannot occur, which yields automatic-continuity results: all derivations and epimorphisms on commutative unital semiprime Banach algebras are continuous, and the separating ideals associated with derivations or epimorphisms are nilpotent. Consequently, derivations on integral-domain and semiprime algebras are continuous, and epimorphisms onto integral domains are continuous as well. The work thus connects prime-ideal structure with automatic continuity phenomena in the Banach-algebra setting and highlights nilpotence of separating ideals as a recurring theme.

Abstract

It is proved that in a commutative unital Banach algebra, every non-maximal closed prime ideal is accessible. Specifically, it can be represented as the intersection of all closed ideals of the algebra that properly contain it. Consequently, all derivations and epimorphisms on commutative unital semi-prime Banach algebras are continuous. Moreover, any separating ideal in a commutative unital Banach algebra is nilpotent and, therefore, a nil ideal.
Paper Structure (4 sections, 12 theorems, 2 equations)

This paper contains 4 sections, 12 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Final Remarks

Key Result

Proposition 2.1

Let $\mathcal{R}$ be a commutative ring with identity, and let $P$ be a non-maximal prime ideal of $\mathcal{R}$. Then either $P$ is equal to the intersection of all prime ideals of $\mathcal{R}$ properly containing $P$, or there exists a sequence $\{ I_n \}$ of nonzero ideals such that $P = \bigcap

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 14 more